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User blog:Rgetar/Ordinals array function
I interested in ordinals, and I asked myself: "Why we use \(\omega\), \(\omega2\), \(\omega^2\), \(\omega^\omega\), \(\epsilon_0\), \(\zeta_0\), \(\phi(3,0)\), \(\Gamma_0\) etc. instead of \(\omega\uparrow\uparrow\uparrow\omega\), \(\omega\uparrow\uparrow\uparrow\uparrow\omega\) etc.?" But I realized this doesn't work beyond \(\epsilon_0\) since \(\omega^{\epsilon_0} = \epsilon_0\). I experimented and created a BEAF-like system, suitable also for ordinals. It is family of functions Xa of ordinal a. Let X - sequence of ordinals (or "negative ordinals", i.e. ordinals with "-" sign before them) with separators. All zeros we may omit. All left part only with zeros and any separators we also may omit. A separator may be written two ways: set of (...) or <...>. A separator has body and type, which are also sequences of ordinals. For example, <\(X_{type}\)|\(X_{body}\)>. If type = 0, separator may be written only with body: <|\(X_{body}\)> = <\(X_{body}\)>. (For now, we'll consider only zero types). (X) is as in BEAF: (1) moves through one row, (2) - through one plane etc. () should move through one element. Easier way is to use : first right element is number of ()'s, second right is number of (1)'s, third right is number of (2)'s etc. In particular, <1> = (X) > = n (X)'s <1> = () may be written as comma (,): <1> = , = () = (0) Examples: <5> = ,,,,, = ,0,0,0,0,0 = ()()()()() = (0)(0)(0)(0)(0) <1,2,3,4,5> = (0)(0)(0)(0)(0)(1)(1)(1)(1)(2)(2)(2)(3)(3)(4) <1,> = <1,0> = <1<1>> = (1) <1,2> = ()()(1) = (0)(0)(1) = ,,(1) = ,0,0(1) <2,> = <2,0> = (1)(1) <1,,> = <1,0,0> = <1<2>> = (2) <1,,,> = <1,0,0,0> = <1<3>> = (3) <2,,3,> = <2,0,3,0> = (1)(1)(1)(3)(3) <3<1,>> = <3<1,0>> = <3<1<1>>> = <3(1)> = (1,)(1,)(1,) = (1,0)(1,0)(1,0) <> = <0> is empty separator, it doesn't add anything: X<> = X<0> = X For example, 1,2,3<> = 1,2,3 <-1> erases one element: X,n<-1> = X For example, 1,2,3<-1> = 1,2 (Also, <-n> should erase n elements, <-n,0> should erase n rows, <-n,0,0> - n planes, but this is not used). X-1 is X with last element decreased by 1. For example, X = 1,2,3,4,5 X-1 = 1,2,3,4,4 is a separator at the right end of X. X* is rest of X. So, X = X* For example, if X = 1,2,3 = 1,2,3<> then X* = 1,2,3 X' = 0 If X = 1(2)2(1)3,0,0,0,0,0 = 1<1<2>>2<1<1>>3<5> = 1<1,,>2<1,>3<5> then X* = 1(2)2(1)3 = 1<1<2>>2<1<1>>3 = 1<1,,>2<1,>3 X' = 5 If X = 1(1)1 = 1<1,0>1<0> = 1<1,>1<> then X* = 1<1,>1 X' = 0 X* and X' are also may be represented in this form: X* = X** X' = X'* Always should be X* = X** = X**<0> = X** X*' = 0 Hence if X = 0 then X* = 0 X' = 0 Any X may be uniquely represented in this form. -1 doesn't change anything: -1a = a X⁰ is X with last element set to zero, if X' = 0 else X⁰ = -1. For example, if X = 1,2,3,4,5 then X⁰ = 1,2,3,4,0 If X = 1,2,3,4,0 then X⁰ = -1 If X = 0 then X⁰ = 0 So, if last element of X is not zero, X⁰ sets it to zero, and if it is zero, X⁰ = -1 except X = 0 when X⁰ also sets it to zero. X·a is a function mapping X to other "X", depending on ordinal a. 0·a = -1 If non-zero type separators not used, rules: 1. []а = a + 1 2. Xa = X⁰X·aa 3. X·a = X*-1<1>a And, I think, rule for limit ordinals, possibly, may be: if last ordinal in X (including ordinals inside separators) is a limit ordinal \(\alpha\), Xa is limit of \(X_i\)a where \(X_i\) are X with \(\alpha\) replaced by \(\alpha_i\) which limit is \(\alpha\). Category:Blog posts